Math
Rørdam K-theory
2.2.10 Liftings
Chapter 2 Projections and Unitary elements
Chapter 3 The K_0 Group of a Unital C -algebra.
Chapter 4 The functor K0
Chapter 5 The ordered abelian group K0
Chapter 6 Inductive limits
Characterization for stable equivalence
Corollary 2.1.4
Definition 2.1.2.
Definition 2.2.1
Definition 2.3.1 (The semigroup of projections of A)
Definition 2.3.3. (The semigroup of projections)
Definition 3.1.4 (The K_0 group of a Unital C-algebra)
Definition 3.1.6 (Stable equivalence)
Definition 3.2.5 (Homotopy equivalence)
Definition 4.1.1 (K0, non-unital case)
Definition 4.2.1 (The Scalar mapping)
Definition 5.1.1.
Definition 5.1.3
Definition 5.1.4 (Positive cone of K0)
Definition 5.16
Definition 6.4.a (Stabilization)
Drawing 2024-02-05 22.07.46
DRAWING
Example 3.1.3
Example 3.3.2
Example 3.3.3
Example 3.3.5
Example 4.1.5
Example 4.3.5
Lemma 2.1.3
Lemma 2.1.5 (Whitehead)
Lemma 2.1.7.
Lemma 2.2.3
Lemma 3.2.7
Lemma 3.2.8
Lemma 4.3.2
Lemma 5.1.2
Lemma 6.3.1
Murray-von Neumann equivalence
Paragraph 3.3.1 Traces and K0
Properties of the Grothendieck group
Proposition 2.1.6.
Proposition 2.1.8
Proposition 2.1.11
Proposition 2.2.5
Proposition 2.2.6
Proposition 2.2.7
Proposition 2.2.8
Proposition 2.3.2.
Proposition 3.1.7 (The standard picture for K0 — the unital case)
Proposition 3.1.8 (Universal property of K0)
Proposition 3.2.4 (Functoriality of K0 for unital C-star Algebras)
Proposition 3.2.6 (Homotopy invariance of K0)
Proposition 4.1.3 (Functoriality of K0)
Proposition 4.1.4 (Homotopy invariance of K0)
Proposition 4.3.2 (Half exactness of K0)
Proposition 4.3.3 (Split exactness of K0)
Proposition 4.3.4 (Direct sums)
Proposition 4.3.8 (Stability)
Proposition 5.1.5
Proposition 5.1.7
Proposition 5.1.9.
Proposition 6.2.4
Proposition 6.4.1 (Stability of K0)
retract
Rørdam K-theory
The functor K00 3.2.3
The functor K0 for unital C-star Algebras 3.2.2
The Grothendieck construction 3.1.1
The group K_00 3.1.5
Theorem 2.1.9. (The polar decomposition)
Theorem 2.1.10 (Carl Neumann Series)
Theorem 4.1.2. (The functor K0)
Theorem 5.2.1
Theorem 5.2.2
Theorem 6.3.2 (Continuity of K0)
Diffuse measure
dirac measures
finite C-star algebra
finite projection
homotopy
infinite C-star algebra
infinite projection
probability measure
quasi-trace
stably finite

Proposition 2.2.6

Let be a C-algebra, let Then in if and only if there exists a unitary element homotopic to the identity which implements unitary equivalence,
$q=upu^{}$.

Proof:
Throughout this proof, denotes the unit of .
Suppose first that for some . Let be a continuous map of unitaries in from to .
Then, since is an ideal of is a continuous path of projections in from to .(what?)
(It is clear that for each t, we have a projection, and the path is the desired one, continuity in P(A) requires A to be an ideal iI guess.)
Conversely, if , then there exists projections in such that for .(How on earth is it the case that these exist?)
It therefore suffices to prove the implication in the case where .
Put then , and Hence is invertible and in by Proposition 2.1.11.
Let be the polar decomposition for , see paragraph 2.19.
Then . From the properties of the polar decomposition we have that inside of , and this entails that belongs to . By Proposition 2.1.8.

Table Of Contents
Proposition 2.2.6